Kaplansky density theorem

Kaplansky density theorem In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books[1] that, The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays. Contents 1 Formal statement 2 Proof 3 See also 4 Notes 5 References Formal statement Let K− denote the strong-operator closure of a set K in B(H), the set of bounded operators on the Hilbert space H, and let (K)1 denote the intersection of K with the unit ball of B(H).

Kaplansky density theorem.[2] If {displaystyle A} is a self-adjoint algebra of operators in {displaystyle B(H)} , then each element {displaystyle a} in the unit ball of the strong-operator closure of {displaystyle A} is in the strong-operator closure of the unit ball of {displaystyle A} . In other words, {displaystyle (A)_{1}^{-}=(A^{-})_{1}} . If {displaystyle h} is a self-adjoint operator in {displaystyle (A^{-})_{1}} , then {displaystyle h} is in the strong-operator closure of the set of self-adjoint operators in {displaystyle (A)_{1}} .

The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.

1) If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ denotes the set of positive operators in A.

2) If A is a C*-algebra acting on the Hilbert space H and u is a unitary operator in A−, then u is in the strong-operator closure of the set of unitary operators in A.

In the density theorem and 1) above, the results also hold if one considers a ball of radius r > 0, instead of the unit ball.

Proof The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {aα} of self-adjoint operators in A, the continuous functional calculus a → f(a) satisfies, {displaystyle lim f(a_{alpha })=f(lim a_{alpha })} in the strong operator topology. This shows that self-adjoint part of the unit ball in A− can be approximated strongly by self-adjoint elements in A. A matrix computation in M2(A) considering the self-adjoint operator with entries 0 on the diagonal and a and a* at the other positions, then removes the self-adjointness restriction and proves the theorem.

See also Jacobson density theorem Notes ^ Pg. 25; Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, ISBN 978-0125494502. ^ Theorem 5.3.5; Richard Kadison, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191. References Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191. V.F.R.Jones von Neumann algebras; incomplete notes from a course. M. Takesaki Theory of Operator Algebras I ISBN 3-540-42248-X hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Von Neumann algebrasTheorems in functional analysis